关于有限群作用于Fredholm算子的Lefschetz不动点型公式的技术问询

Hey everyone, let's start by laying out some foundational definitions to frame this technical discussion:

• Let $D:X\to Y$ be a Fredholm operator, meaning its index $\dim\ker(D) - \dim\text{coker}(D)$ is well-defined.
• We can interpret $D$ as a chain complex $\mathfrak{D}:0\to X\xrightarrow{D} Y\to 0$. For a chain map $T:\mathfrak{D}\to\mathfrak{D}$ — which corresponds to the commutative diagram below:
  $$
  \begin{CD}
  X @>{D}>> Y\
  @V{T}VV @V{T}VV\
  X @>{D}>> Y
  \end{CD}
  $$
• This chain map induces two linear maps: $T_0:\ker(D)\to\ker(D)$ (acting on the kernel of $D$) and $T_1:\text{coker}(D)\to\text{coker}(D)$ (acting on the cokernel of $D$).
• The Lefschetz number $L(T)$ of $T$ is then defined as the difference of the traces of these induced maps: $L(T)=\text{trace}(T_0)-\text{trace}(T_1)$.

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备注：内容来源于stack exchange，提问作者user302934